# Analogues of the Balog--Wooley Decomposition for Subsets of Finite Fields and Character Sums with Convolutions

## TL;DR

This paper extends the Balog--Wooley decomposition to finite fields, showing how subsets can be split into parts with small additive and multiplicative energies, and applies this to estimate complex character sums involving convolutions.

## Contribution

It provides an analogue of the Balog--Wooley decomposition for finite fields under certain conditions and introduces bounds on character sums involving convolutions with rational functions.

## Key findings

- Decomposition into parts with small additive and multiplicative energies in finite fields.
- Bounds on character sums involving convolutions of subsets.
- Optimal bounds up to logarithmic factors for subset sizes.

## Abstract

Balog and Wooley have recently proved that any subset $A$ of either real numbers or of a prime finite field can be decomposed into two parts $U$ and $V$, one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function $f$ both the additive energies of $U$ and $f(V)$ are small. Our method is based on bounds of character sums which leads to the restriction $\# A > q^{1/2}$ where $q$ is the field size. The bound is optimal, up to logarithmic factors, when $\# A \geq q^{9/13}$. Using $f(X)=X^{-1}$ we apply this result to estimate some triple additive and multiplicative character sums involving three sets with convolutions $ab+ac+bc$ with variables $a,b,c$ running through three arbitrary subsets of a finite field.